Timeline for Continuous point map for spherical domains
Current License: CC BY-SA 4.0
26 events
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| Aug 8, 2023 at 12:56 | answer | added | Mohammad Ghomi | timeline score: 1 | |
| Jul 24, 2023 at 19:30 | history | edited | Mohammad Ghomi |
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| Jul 20, 2023 at 13:19 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 20, 2023 at 12:37 | history | edited | Mohammad Ghomi |
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| Jul 20, 2023 at 12:07 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 20, 2023 at 11:56 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 20, 2023 at 11:42 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 20, 2023 at 8:10 | comment | added | Geoffrey Irving | Oops, of course: I was hallucinating that it was unique up to rotation, but it’s unique up to rotation and recentering, which is useless. | |
| Jul 19, 2023 at 23:14 | comment | added | Martin M. W. | @GeoffreyIrving Is that really well-defined, since the Riemann mapping isn't unique? | |
| Jul 19, 2023 at 20:12 | comment | added | Geoffrey Irving | I think it does work, because if two sets are close, there is a small shrinking of both sets such that the Riemann map to the common shrunk set is close to the Riemann map to each set. | |
| Jul 19, 2023 at 20:07 | comment | added | Geoffrey Irving | Does the image of the origin under a Riemann mapping of the disk to the set work? This is related to some the above and below proposals, but I can’t tell if it’s the same. | |
| Jul 18, 2023 at 22:32 | answer | added | Ian Agol | timeline score: 3 | |
| Jul 14, 2023 at 15:07 | comment | added | Mohammad Ghomi | @Alexandre Eremenko, the topology is already defined as the space of continuous injective maps from the disk to the sphere modulo homeomorphisms of the disk. So two domains are close provided that they admit parameterizations which are $C^0$ close. This is different from Hausdorff topology, where as you say bad things happen. | |
| Jul 14, 2023 at 14:03 | comment | added | Alexandre Eremenko | To define continuity of your f, you need to specify topology on J. What is it? Convergence defined by Hausdorff metric? You can have all sorts of weird limit in such topology. | |
| Jul 14, 2023 at 11:38 | comment | added | RBega2 | Yes, as I alluded to CSF will give you a pair of antipodal points (the singular point and its antipodal point or the two antipodal focal points of the geodesic) and you have to use the side the disk is on to select the ``correct" one. I imagine this relates to a topological result about what happens when the curves are oriented or unoriented. | |
| Jul 14, 2023 at 2:38 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 14, 2023 at 2:37 | comment | added | Mohammad Ghomi | @RBega, I don't think that CSF is going to work even for smooth curves, because if we take a pair of smaller circles on either side of a great circle and parallel to it they will float towards antipodal points. | |
| Jul 14, 2023 at 1:26 | comment | added | RBega2 | I guess the paper I linked required the image of the curve to have Lebesgue measure zero in order to have a unique evolution, though I'm an not totally sure how you define center of mass in a continuous fashion in this case either. | |
| Jul 14, 2023 at 1:22 | comment | added | RBega2 | There is a well defined CSF for Jordan curves in $\mathbb{S}^2$. Not sure about making it area preserving. Though it might be possible to either use $\pm$ the extinction point or the center if the limit was a geodesic. | |
| Jul 14, 2023 at 1:14 | comment | added | Mohammad Ghomi | @RBega, as far as I know curve shortening flow would require the curve to be rectifiable. Also I do not know how you would keep the area constant. | |
| Jul 14, 2023 at 1:06 | comment | added | RBega2 | Presumably you could run area preserving curve shortening flow which should converge to a circle in the geodesic circle in $S^2$ and then map to the center of the circle (with the ambiguity of which choice to make determined by the domain). The only part that is not totally clear to me is the continuity, but that seems plausible. This may be overkill of course. | |
| Jul 14, 2023 at 0:02 | comment | added | Marco Golla | I naively thought this could not happen---I can envision some counterexamples now. | |
| Jul 14, 2023 at 0:00 | comment | added | Mohammad Ghomi | @Marco Golla, it does not work when the center of mass is at the origin. | |
| Jul 13, 2023 at 23:59 | comment | added | Marco Golla | What if you take the centre of mass in $\mathbb{R}^3$ and project radially? | |
| Jul 13, 2023 at 23:59 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Jul 13, 2023 at 23:49 | history | asked | Mohammad Ghomi | CC BY-SA 4.0 |